On Bloch waves for the stokes equations

Grégoire Allaire; Carlos Conca; Luis Friz; Jaime H. Ortega

doi:10.3934/dcdsb.2007.7.1

On Bloch waves for the stokes equations

Grégoire Allaire
, Carlos Conca
, Luis Friz
, Jaime H. Ortega

Research output: Contribution to journal › Article › peer-review

Abstract

In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in ℝ^d. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency ξ, are not continuous at the origin. Nevertheless, when ξ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.

Original language	English
Pages (from-to)	1-28
Number of pages	28
Journal	Discrete and Continuous Dynamical Systems - Series B
Volume	7
Issue number	1
DOIs	https://doi.org/10.3934/dcdsb.2007.7.1
Publication status	Published - 1 Jan 2007

Keywords

Bloch waves
Homogenization
Spectral theory
Stokes equation

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10.3934/dcdsb.2007.7.1

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abstract = "In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in ℝd. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency ξ, are not continuous at the origin. Nevertheless, when ξ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.",

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AU - Allaire, Grégoire

AU - Conca, Carlos

AU - Friz, Luis

AU - Ortega, Jaime H.

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Y1 - 2007/1/1

N2 - In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in ℝd. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency ξ, are not continuous at the origin. Nevertheless, when ξ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.

AB - In this work, we study the Bloch wave decomposition for the Stokes equations in a periodic media in ℝd. We prove that, because of the incompressibility constraint, the Bloch eigenvalues, as functions of the Bloch frequency ξ, are not continuous at the origin. Nevertheless, when ξ goes to zero in a fixed direction, we exhibit a new limit spectral problem for which the eigenvalues are directionally differentiable. Finally, we present an analogous study for the Bloch wave decomposition for a periodic perforated domain.

KW - Bloch waves

KW - Homogenization

KW - Spectral theory

KW - Stokes equation

UR - https://www.scopus.com/pages/publications/33847749831

U2 - 10.3934/dcdsb.2007.7.1

DO - 10.3934/dcdsb.2007.7.1

M3 - Article

AN - SCOPUS:33847749831

SN - 1531-3492

VL - 7

SP - 1

EP - 28

JO - Discrete and Continuous Dynamical Systems - Series B

JF - Discrete and Continuous Dynamical Systems - Series B

IS - 1

ER -

On Bloch waves for the stokes equations

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Keywords

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